How to modify the Langlands' dual group

Let $\mathcal G$ be a split reductive group over a $p$-adic field $F$, and $G$ the group of its $F$-points.

The main insight of the local Langlands program is that to every irreducible smooth representation $(\rho, G, V )$ should correspond a morphism $\nu_\rho : W_F \to {}^\vee G$ of the Weil group $W_F$ of the field $F$ to the Langlands' dual group $^\vee G$.

Ideally this should extend to a bijection between the set $Irr(G)$ of irreducible representations of $G$ and some set $Lan({}^\vee G)$ of Langlands parameters described in terms of the dual group. Even better would be to describe the category $Rep(G)$ of smooth $G$-modules in terms of the dual group.

By now it is clear that to achieve these goals one should modify both the left hand side $Rep(G)$ and the right hand side.

In my talk I will try to explain that in fact, starting from the first principles, it is clear that we should work not with the Langlands' dual group ${}^\vee G$ but with some slight modification of it.

For example, in case of the group $G = PGL(2,F)$ one should assign to an irreducible representation $\rho$ of the group $G$ a morphism
\[ \nu = \nu_\rho : W_F \to GL(2, \mathbf C) \]
that satisfies the condition $\det(\nu) = \omega$, where $\omega : W_F \to \mathbf C^∗$ is the cyclotomic character.

I will try to describe this modification and discuss possible formulations of Local Langlands correspondence.

I will also discuss how this modification affects the notion of $L$-functions corresponding to an automorphic representation $\pi$. In particular, we will see that in this approach it is quite clear where to look for the special values of these $L$-functions.